Which Of The Following Are Solutions To The Equation Below? Check All That Apply. 3 X 2 + 27 X + 60 = 0 3x^2 + 27x + 60 = 0 3 X 2 + 27 X + 60 = 0 A. -4 B. -5 C. -27 D. 5 E. 4

Mar 10, 2025 by ADMIN 179 views

**Solving Quadratic Equations: A Step-by-Step Guide**

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore the solutions to the quadratic equation $3x^2 + 27x + 60 = 0$ and determine which of the given options are correct.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. In our given equation, $a = 3$ , $b = 27$ , and $c = 60$ .

The Quadratic Formula

To solve quadratic equations, we can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . This formula provides two solutions for the equation, which are the values of $x$ that satisfy the equation.

Applying the Quadratic Formula

Let's apply the quadratic formula to our given equation: $3x^2 + 27x + 60 = 0$ . We have $a = 3$ , $b = 27$ , and $c = 60$ . Plugging these values into the quadratic formula, we get:

$x = \frac{-27 \pm \sqrt{27^2 - 4(3)(60)}}{2(3)}$

$x = \frac{-27 \pm \sqrt{729 - 720}}{6}$

$x = \frac{-27 \pm \sqrt{9}}{6}$

$x = \frac{-27 \pm 3}{6}$

Simplifying the Solutions

Now, we have two possible solutions:

$x = \frac{-27 + 3}{6}$

$x = \frac{-24}{6}$

$x = -4$

$x = \frac{-27 - 3}{6}$

$x = \frac{-30}{6}$

$x = -5$

Evaluating the Options

Now that we have the solutions to the equation, let's evaluate the given options:

A. -4 B. -5 C. -27 D. 5 E. 4

Based on our calculations, options A and B are correct solutions to the equation.

Conclusion

In this article, we solved the quadratic equation $3x^2 + 27x + 60 = 0$ using the quadratic formula and determined which of the given options are correct solutions. We found that options A and B, -4 and -5, respectively, are the solutions to the equation.

Final Thoughts

Solving quadratic equations is an essential skill in mathematics, and understanding the quadratic formula is crucial for solving these types of equations. By applying the quadratic formula and simplifying the solutions, we can determine the correct solutions to the equation. In this case, options A and B are the correct solutions to the equation.

Additional Resources

For more information on quadratic equations and the quadratic formula, check out the following resources:

Khan Academy: Quadratic Equations
Mathway: Quadratic Formula
Wolfram Alpha: Quadratic Equation Solver

Practice Problems

Try solving the following quadratic equations using the quadratic formula:

$x^2 + 5x + 6 = 0$
$2x^2 + 3x - 1 = 0$
$x^2 - 4x + 4 = 0$

Glossary

Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable is two.
Quadratic Formula: A formula used to solve quadratic equations, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Solutions: The values of $x$ that satisfy the equation.

References

"Algebra and Trigonometry" by Michael Sullivan
"College Algebra" by James Stewart
"Quadratic Equations" by Khan Academy

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding them is crucial for solving a wide range of problems. In this article, we will answer some of the most frequently asked questions about quadratic equations, providing a comprehensive guide to help you better understand this topic.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . This formula provides two solutions for the equation, which are the values of $x$ that satisfy the equation.

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. For example, if you have the equation $x^2 + 5x + 6 = 0$ , you would plug in $a = 1$ , $b = 5$ , and $c = 6$ into the formula.

Q: What is the difference between the two solutions provided by the quadratic formula?

A: The two solutions provided by the quadratic formula are the values of $x$ that satisfy the equation. The difference between the two solutions is the sign of the square root term. If the square root term is positive, the solutions will be real numbers. If the square root term is negative, the solutions will be complex numbers.

Q: Can I use the quadratic formula to solve all quadratic equations?

A: Yes, the quadratic formula can be used to solve all quadratic equations. However, it's worth noting that the quadratic formula may not always provide a simple or straightforward solution. In some cases, the solutions may be complex numbers or irrational numbers.

Q: How do I determine if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex coefficients. However, you will need to use complex numbers and complex arithmetic to solve the equation.

Q: How do I simplify the solutions provided by the quadratic formula?

A: To simplify the solutions provided by the quadratic formula, you can use algebraic manipulations such as factoring, combining like terms, and canceling common factors.

Q: Can I use the quadratic formula to solve quadratic equations with rational coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with rational coefficients. However, you will need to use rational arithmetic to solve the equation.

Q: How do I determine if a quadratic equation has rational or irrational solutions?

A: To determine if a quadratic equation has rational or irrational solutions, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is a perfect square, the equation has rational solutions. If the discriminant is not a perfect square, the equation has irrational solutions.